In the oil and gas industry, a technique called tomographic inversion is used to build models in the form of data volumes giving seismic wave velocity values within a subsurface region of interest. Such techniques take several forms.
1. Uniform Grid Tomographic Inversion with Smoothing Constraint
An approach considered state-of-the-art uses a uniform grid (e.g. a grid consisting of rectangular cells) to parameterize the model and tries to stabilize the inversion by adding smoothing constraints. In general, smoothing constraints are effective in removing short-wavelength artifacts but fail to remove long-wavelength artifacts. Smoothing will also fail if the artifacts in one area are of the same wavelength as true velocity structure elsewhere in the model.
2. Multi-Scale Tomographic Inversion
This approach uses a multi-scale of uniform grids to parameterize a model. The lower scale grid model has longer wavelength and, therefore, yields a more stable solution. The multi-scale grid models can be solved progressively or simultaneously. This method multiplies the computational cost in ray tracing by the number of scales. In addition, this method still has the instability due to the underdetermined model parameters and will fail if artifacts in one area are of the same wavelength as true velocity structure elsewhere in the model.
3. Non-Uniform Grid Tomographic Inversion
This approach uses a non-uniform grid to parameterize a model. The non-uniform parameterization generally has many fewer parameters than a uniform grid. In this way, the tomographic inversion is stabilized by reducing the null space of the matrix operator. However, ray tracing is generally much slower in a model parameterized with a non-uniform grid. Furthermore, it is hard to add smoothing constraints to a non-uniform grid model.